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Higher Moments Risk and the Cross-Section of Stock Returns

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Higher Moments Risk and the Cross-Section of Stock Returns HIGHER MOMENTS RISK AND THE CROSS-SECTION OF STOCK RETURNS Garazi Elorza Iglesias Trabajo de investigación 016/007 Master en Banca y Finanzas Cuantitativas Tutores: Dr. Alfonso Novales Universidad Complutense de Madrid Universidad del País Vasco Universidad de Valencia Universidad de Castilla-La Mancha www.finanzascuantitativas.com Higher moments risk and the cross-section of stock returns Garazi Elorza Director: Alfonso Novales M´aster en Banca y Finanzas Cuantitativas Madrid Contents 1 Introduction 4 2 Data 6 3 The analytical framework 6 4 Market moments innovations 8 4.1 Estimation of higher moments . .8 4.1.1 GARCHSK . .9 4.1.2 NAGARCHSK . .9 4.2 Innovations in market moments . 12 5 Portfolios sorted on market moments 15 5.1 Portfolios sorted on market volatility exposure . 15 5.2 Portfolios sorted on market skewness exposure . 17 5.3 Portfolios sorted on market kurtosis exposure . 18 5.4 NAGARCHSK model . 21 5.5 Results on subperiods . 21 5.6 Using rolling window . 22 6 Factor portfolios 23 7 Exploring the risk premiums 26 7.1 Fama and MacBeth regressions on the 81 factor portfolios . 27 7.2 Fama and MacBeth regressions on the other portfolios . 30 7.3 Interpreting the sign of the price of market moments risk . 30 8 Sorting stock returns on different moments 33 9 Conclusion 35 A Appendix 1 38 B Appendix 2 39 B.1 Autocorrelation functions of market moments with GARCHSK . 39 C Appendix 3 40 C.1 Results for 10 portfolios . 40 2 D Appendix 4 41 D.1 Results for NAGARCHSK model . 41 E Appendix 5 43 E.1 Results for rolling window . 43 3 4 1 Introduction It is well known that excess kurtosis and negative skewness are standard characteristics of the stock return distributions. So it seems natural to take them into the account in asset pricing. The presence of the excess kurtosis in an index means that the market gives more probability to extreme observations than in normal distribution. Meanwhile, the appearance of negative skewness has the effect of highlighting the left tail of the distribution. In that case, market gives higher probability to decreases than increases in asset pricing. It is by now well accepted that market skewness and kurtosis are important indicators of market risk, so the goal of this thesis is twofold: i) to analyze whether market skewness and kurtosis risks affect to the cross-section of stock returns, ii) to examine whether individual skewness and kurtosis are priced in the market. We use daily closing prices from the Eurostoxx market for which, to the best of our knoweledge, this analysis has not been done. The estimation method for higher moments is based on Gram-Charlier series expansion of the normal density function for the error term, as in Le´onet al. (2004), where GARCH-type models are used allowing for time-varying volatility, skewness and kurtosis. These authors find a significant presence of conditional skewness and kurtosis in daily returns of stock indices and exchange rates. We use a different estimation method for individual stocks, because estimating the parameters of the Exponentially weighted moving average or GARCH models for each stock has a high compu- tational cost. Thus, we decide to calculate higher moments series for each stock as in RiskMetrics imposing a parameter λ = 0:94 for all the cases. Nevertheless, there are many methods for calculating higher moments. For example, recent studies by Chang et al. (2011) and Bams et al. (2015) show that such moments could be calculated using out-of-the-money European call and put options prices. High-frequency returns of a single day can also be used to compute the moments in a particular day as in Amaya et al. (2015). We could use the traditional technique of rolling window of daily returns. We perform two types of empirical exercises to analyze if market higher moments risks affect to stocks' risk premium. First, we sort all stocks in Eurostoxx from 2000 to 2016 in quintiles based on their exposure of their returns to each moment's innovation, as in Chang et al. (2011) and Bams et al. (2015). These authors find that in down-markets, when investors are more risk-averse, the market volatility risk is priced significantly negative, while the effect disappears in up-markets. As for higher moments risk premium, skewness and kurtosis, they conclude that the risk premium for these moments are significantly negative and positive, respectively, but only when the investors risk aversion is low. Similar results can be seen in Chang et al. (2011), where they also show that stocks with high exposure to innovations in market skewness have low returns on average. Nonetheless, the results are weaker for volatility and kurtosis, where stocks with high exposure in market volatility and 5 kurtosis exhibit somewhat lower and higher returns on average, respectively. We can find in the literature different techniques for portfolio construction: either equal- weighted portfolios, which are portfolios where all the stocks have the same weight, or value- weighted portfolios that are constructed according to the value they have in relation to the total of the portfolio. We use equal-weighted portfolios, and we find some evidence that market volatility is priced in the cross section of stocks, where stocks with high exposure to innovations in market volatility risk exhibit low returns on average. The results for market skewness and kurtosis are not so clear, but they show that the stocks with high exposure to these moments exhibit higher returns on average. Therefore, factor portfolios for market volatility, skewness and kurtosis risk are constructed. For that, firstly, all stocks of Eurostoxx are sorted in terciles based on their exposure of their returns to innovations in either market moment (index return, volatility, skewness and kurtosis) and then these 12 groups are combined. We obtain that the average return on the market skewness and kurtosis risk factor portfolio are −0:10% and −0:08% per month, respectively, or −1:20% and −0; 98% per year. As a second approach, the prices of the market moment risks are also estimated, using Fama and MacBeth regressions as in Fama and MacBeth (1973). This two-step estimation approach is very popular. Amaya et al. (2015) use it to determine the significance on the cross-section of the stock returns of each market high moment individually and jointly. Chang et al. (2011) use this methodology to estimate the price of market high moments risks. As in that work, we study whether market higher moments risks affect to the cross-section of stock returns and we use Fama and MacBeth methodology for calculating risk premiums. We find that the estimates of the premium for market volatility and kurtosis risk are negative and for market skewness risk is positive. The remainder of the work is organized as follows. In section 2 the data is discussed. In section 3 the models used for computing the market moments' risk premia are discussed. Section 4 presents the methods used to extract higher moments from Eurostoxx index returns, as well as the extraction of the innovations. In section 5, there are presented the results for the stocks that are sorted into quintiles based on their exposure to innovations in market moments. Section 6 constructs factor portfolios and in section 7 estimates the price of market moments risk using Fama and MacBeth regressions. Section 8 presents the results for portfolios sorted on realized moments. Section 9 concludes. 6 2 Data We use individual stocks from Eurostoxx which is a stock index of the Eurozone and it is com- pounded by 293 important firms. The data set includes daily closing prices from January 3, 2000 to April 7, 2016. If these period is considered there are just 203 companies because some of them are newer and enter in the index later. We also need the Eurostoxx index prices and they are obtained from DataStream, as well as the stocks data. The factor mimicking portfolio returns for size, book-to-market and momentum factors are obtained from the online data library of Ken French which can be found at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ data_library.html. On the other hand, we also need the return of the risk-free asset, which is considered German 10-year bond and it is also obtained from Datastream. The study is going to focus on stock returns rather than with prices, so given a series of asset prices P0;P1; ··· ;PT continuously compounded returns for period t are defined as Rt = ln(Pt=Pt−1), t = 1; 2;:::;T . 3 The analytical framework In this study, there are certain risk factors, which are market higher moments, namely volatility, skewness and kurtosis and the goal is to study empirically their effect to the cross section of stock returns. We use two strategies to investigate this and we use a multifactor representation of equilibrium returns, with the moments of the market return as state variables. The first strategy is based on univariate sorting. The factors are moments of market return. So if we use a sample of returns and moments for a period t = 1; 2;:::;T , the model is defined as: Rit − Rf;t = βi0 + βi;MKT (Rm;t − Rf;t) + βi;∆V ol∆V olt + βi;∆Skew∆Skewt + βi;∆Kurt∆Kurtt + it (3.1) where Rit is the ith risky asset return, Rf;t is the return of the risk-free asset and Rm;t the market portfolio return. Furthermore, ∆V olt = V olt − Et−1[V olt], ∆Skewt = Skewt − Et−1[Skewt] and ∆Kurtt = Kurtt − Et−1[Kurtt].
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